I have a homework questions that I am stuck on, it's the last in the set and I can't do it!

It reads:

Prove that the meridians are geodesics when the curve: (x(u),0,u) is revolved around the z-axis.

I have an idea of using the Geodesic Curvature formula etc but do not know where to start.

If anyone could give me a hand, please let me know!

Cheers.

Patrick.

Oct 2nd 2012, 06:54 AM

johnsomeone

Re: Meridians onf surface of revolution.

Consider a constant speed curve along a meridian of the revolved surface S,
where is some open interval of .
To show is a geodesic, you want to show that is orthogonal to S.

Since it has constant speed, you can say something about the relationship between and .
Consider the plane P that defines the meridian by cutting through S.
Since stays in P, so do all of its derivatives.
At any point along , P is spanned by and the normal to S at that point.
Put it all together, and you deduce something about that proves that is a geodesic.